# Who said $\pi$ is a constant since it is not even a real number?

EDIT: (130116) I don't mean it is complex or imaginary nor it is negative also, I tried hard to conceive it on the real line number (positive X-axis), by obvious means, a little idea came to me?, "every time you locate it between two digits, enlarge your number line say 10 times (in base 10 counting)", in order to locate it more clearly, after many finite digits I got tired, and also noticed that my number line is becoming longer than our milky way galaxy, then I simply realized I will never get it on our number line, because simply it is not there on our number line even at infinity?, it is a kind of distinct infinity that is not there?, it is simply an "illusion"

It is by definition $\pi$ is a ratio of circumstance to diameter of a circle, this was defined thousands of years back where the concept of numbers may had been restricted mainly on rational numbers, they could approximate it to few digits, and since then with the help of supercomputers nowadays it was approximated to many trillion digits, and we all know that $\pi$ never ends (being irrational number that is impossible to construct), it is always a ratio of two integers say $n/m$, where both $n$ & $m$ don't exist, then how can we consider it as a constant?

It is rather a non existing number on our real number line, because first it is impossible to construct (with rigorous proof), second there is no circle with exactly an integer diameter & integer circumstance (also this is proved rigorously), that doesn't necessarily imply its non existence or consistency in other "undefined" number line!?

• @MauroALLEGRANZA Please note that questions about $\pi$ are numerous, just give it a chance, they are different – Bassam Karzeddin Dec 19 '15 at 13:52
• You can see this post for evidence of ancient Greek math regarding the fact that the ratio circumference/diameter is constant. – Mauro ALLEGRANZA Dec 19 '15 at 14:06
• This question should have upvotes – Konstantinos Gaitanas Dec 21 '15 at 8:36
• @bassamkarzeddin, I will provide a separate answer. – Mikhail Katz Dec 21 '15 at 13:13
• This question does not seem to be about anything historical relating to $\pi$. – HDE 226868 Jan 15 '16 at 0:23

This question is based on a misunderstanding. The statement that $\pi$ is constant has precise meaning: $\pi$ is a ratio of the length of circumference to the length of diameter. The statement that it is constant means that it is the same for all circles. (This statement is independent of the representation of this ratio with digits). Contrary to what many people think, the Greeks had a proof of this. The proof is valid, and requires no modification. (The Greeks did not define the general notion of length, but in the case of circle their definition is consistent with the later general definition). By a number, the Greeks understood only integer or rational number, so from their point of view $\pi$ is not a number. But they had a completely rigorous notion of proportion which played the same role, and which is equivalent to the modern theory of real numbers.

Representation of real numbers by infinite sequences of digits is a later invention. But it has nothing to do with the Greek theorem that the ratio of circumference to diameter is the same for all circles.

• The Greeks tried to represent $\pi$ as a fraction, where they found approximation fractions, which is the same as decimal representation, but more closer approximation today, they also tried to construct it, but, they proved the construction of $\pi$ is impossible, if I would think about this matter, we are closer to nothing of what they had done, only closer fractions we can add with each new digit added, but we know this is a meaning less to the content of $\pi$ since the fractions are ENDLESS, but becoming as a continuous game, $\pi$ is not constant or anywhere except in our minds, – Bassam Karzeddin Dec 19 '15 at 16:34
• Yes, they tried to represent (approximate) $\pi$ as a fraction. They also tried to construct it with a compass. They DID NOT know that this is impossible, this is a modern achievement. But independently of this, they had a proof that it does not depend on a circle, that is a constant. – Alexandre Eremenko Dec 19 '15 at 18:42
• To summarize: A number can be a constant without it being known what that constant is. Such is the case with $\pi$. It can be show that for any two circles the ratio of circumference to diameter is the same. That is sufficient to say $\pi$ is constant. And the Greeks knew this, and proved it. – Floris Dec 21 '15 at 3:29
• @AlexandreEremenko Squaring the circle is impossible, and this is surely not a modern mathematics, it is actually a very old problem, which means the ancient knew what modern mathematics had reconfirmed, about $\pi$ being independent of a circle seems ridiculous or at least needs investigation!? – Bassam Karzeddin Dec 21 '15 at 10:13
• @bassamkarzeddin a "ghost" number? Just because we can't write it down with a finite number of digits from 0-9 ? You are confusing "constant" (your question) with "known number". – Floris Dec 21 '15 at 12:47

There are various explicit formulas for $\pi$ that allow you to compute it as precisely as you wish. One of the first ones was provided by Wallis already in the 17th century. I am not sure what exactly is bothering you about $\pi$ but if it is the fact that $\pi$ is irrational, then the problem already exists for simpler numbers like $\sqrt{2}$, namely the diagonal of the unit square. It is hard to deny that such a number should exist as it has a clear geometric counterpart.

Note 1. To respond to a comment of the OP, I don't fully understand your distinction between $\pi$ and $\sqrt{2}$, since $\pi$ is also a constructible number (though it cannot be constructed with ruler and compass) in a purely mathematical sense of constructible, but there is a difference between them in that $\pi$ is transcendental whereas $\sqrt{2}$ is algebraic. Since even a rational number like $\frac{1}{3}$ cannot be represented by finitely many decimal digits, what is exactly your objection to unending decimal representations of $\pi$ and $\sqrt{2}$?

• I replied within the body of the answer. – Mikhail Katz Dec 21 '15 at 14:07
• Leaving aside for a moment the issue of 0.999... and 1, would you maintain that the number $\frac{1}{3}$ does not exist because it requires infinitely many digits? – Mikhail Katz Dec 21 '15 at 14:30
• So you are looking for finite representations for a number? Well, $\pi$ also has a finite representation: it is the area of a unit circle. Why should we say that $\pi$ does not exist? – Mikhail Katz Dec 21 '15 at 14:39
• You seem to think that the geometric description is not enough to define a number. But $\sqrt{2}$ also has a geometric origin: it is the diagonal of a unit square. So why should $\sqrt{2}$ be declared to exist but $\pi$ not to exist? – Mikhail Katz Dec 21 '15 at 14:46
• @bassamkarzeddin, what you are pointing out that $\pi$ is not contained in any pythagorean field, and that is quite correct. If your criterion of existence is constructibility by ruler and compass, then indeed one will not be able to solve the cubic equation. You should consider, however, that this choice of the notion of constructibility is a bit artificial. One could limit oneself to a smaller collection of numbers, such as the rationals, or a larger one, such as the algebraic ones. Ultimately all irrational numbers, including $\sqrt{2}$, are problematic because our instruments are imperfect – Mikhail Katz Dec 21 '15 at 15:11

Following up on Eremenko, see David Richeson's lead article in the May 2015 College Mathematics Journal, Circular reasoning: Who first proved that C divided by d is a constant?

http://www.maa.org/press/periodicals/college-mathematics-journal/college-mathematics-journal-contents-may-2015

Nobody said π is a constant or nobody has to. Like every other number is a constant even π is. Only it's transcendence makes it unique. We knew, as you've mentioned, since the early stages of mathematics about π. But because of the inability to calculate the billions of decimal places of π it was rounded to 22/7.

It's also unclear as to what you are asking. Please try and edit it or at least make me understand.

• Actually, we don't have to be able to find more few trillions of its digits, the only thing we have to show that it is actually not there, anywhere, it is simply an illusionary number that never exists, but still we needs its shadow!? – Bassam Karzeddin Dec 19 '15 at 13:58
• Don't we have shadowban on this site...? – VicAche Dec 20 '15 at 23:21